Is the arrow space of a fundamental groupoid of a Hausdorff and second countable manifold always Hausdorff and second countable?

Question

Usually in the definition of a Lie groupoid, we do not assume the arrow space to be second countable and Hausdorff.

Now, in particular for a smooth manifold $M$ with the assumption that it is second countable and Hausdorff, is the arrow space of $\Pi(M)$ always Hausdorff and second countable? Any reference in this direction is appreciated.